Fig. 31.1 shows a domestic hot water tank being filled with cold water at 12 °C from the mains. The tank has a capacity of 0.15 m³. The density of water is ...

(a) Mass of water in the tank [1]

The relationship between mass, density and volume is:

\( m = \rho V \)

Substituting the given values (density of water = 1000 kg/m³, volume of tank = 0.15 m³):

\( m = 1000 \times 0.15 = \mathbf{150 \text{ kg}} \) [1]

(b) Thermal energy needed to heat the water [2]

The thermal energy required to raise the temperature of a substance is given by:

\( E = mc\Delta\theta \)

The temperature rise is \( \Delta\theta = 60 - 12 = 48 \)°C. Substituting:

\( E = 150 \times 4200 \times 48 \) [1]

\( E = 30\,240\,000 \text{ J} \approx 3.02 \times 10^7 \text{ J} \) [1]

This is a very large amount of energy, which is why heating a full tank of water takes a long time and is expensive.

(c) Minimum time for the heater to deliver this energy [2]

Power is the rate of energy transfer, so \( P = E / t \), which rearranges to:

\( t = \frac{E}{P} = \frac{30\,240\,000}{3000} \) [1]

\( t = 10\,080 \text{ s} = 168 \text{ min} \approx 2.8 \text{ hours} \) [1]

This is the minimum time because it assumes all the electrical energy goes into heating the water with zero losses.

(d) Why the actual heating time is longer [1]

Some thermal energy is lost through the insulation to the surroundings by conduction, convection and radiation. [1] The heater must therefore supply more energy than the calculated value to achieve the desired temperature rise, which takes additional time. The lagging reduces but does not eliminate these losses.

(e) Why the heater element is near the bottom of the tank [2]

Hot water is less dense than cold water, so heated water near the element rises to the top of the tank by convection. [1] Placing the heater at the bottom ensures that as warm water rises, cooler water sinks to replace it and gets heated in turn. This creates a convection current that circulates all the water past the heater, ensuring the entire tank is heated evenly. [1] If the heater were at the top, only the water above it would be heated directly, and the colder, denser water below would remain largely unheated.